Question

Let $BC$ be a fixed line segment in the plane. The locus of a point $A$ such that the triangle $ABC$ is isosceles, is (with finitely many possible exceptional points)

• A. a line
• B. a circle
• C. the union of a circle and a line
• D. the union of two circles and a line

Answer 1

The correct option is D the union of two circles and a line

$BC$ is a fixed line segment in the plane and $A$ is a point such that $△ABC$ is isosceles.

Case I : If $\angle B=\angle C$
Locus of $A$ is perpendicular bisector of $BC$
So, it is a straight line.

Case II : If $\angle A=\angle C,BC$ is fixed.
$BC=AB\Rightarrow$ Locus of point $A$ is a circle.

Case III : If $\angle A=\angle B,BC$ is fixed.
$AC=BC\Rightarrow$ Locus of point $A$ is also a circle.
Hence, locus of point $A$ is union of two circles and a line.